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On definably proper maps

Volume 233 / 2016

Mário J. Edmundo, Marcello Mamino, Luca Prelli, Fundamenta Mathematicae 233 (2016), 1-36 MSC: Primary 03C64; Secondary 55N30. DOI: 10.4064/fm96-12-2015 Published online: 2 December 2015


In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is a proper morphism in the category of definable spaces. We give several other characterizations of definably proper, including one involving the existence of limits of definable types. We also prove the basic properties of definably proper maps and the invariance of definably proper (and definably compact) in elementary extensions and o-minimal expansions.


  • Mário J. EdmundoUniversidade Aberta
    Rua Braamcamp 90
    1250-052 Lisboa, Portugal
    CMAF Universidade de Lisboa
    Av. Prof. Gama Pinto 2
    1649-003 Lisboa, Portugal
  • Marcello MaminoLaboratoire d'Informatique
    de l'École Polytechnique (LIX)
    Bâtiment Turing, bureau 2011
    1 rue Honoré d'Estienne d'Orves
    Campus de l'École Polytechnique
    91120 Palaiseau, France
  • Luca PrelliCMAF Universidade de Lisboa
    Av. Prof. Gama Pinto 2
    1649-003 Lisboa, Portugal

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